beta 2.8 screen capture
In Blender 2.8x, every menu item that doesn't have a shortcut can be assigned one very easily: 


right click on the menu item, then
either select 


Add to Quick Favorites: your quick favorites will be available under theQ shortcut for quick selection
Assign Shortcut: you can now assign a custom unique shortcut.



I think the developers have made sure to leave F4 to F10 unassigned ("minimal keymap" philosophy) so that you can use these keys as custom hotkeys.